Differentiation Questions and Answers
Calculus
DifferentiationLet F x 1 lim F 2 lim F 2 and I 1 I 1 both 2 Find the following limits O both 1 O2 and 1 O2 and 2
Calculus
DifferentiationFind f x and f x of the function f x 5 cos x sin x Select 2 correct answer s Of x 5 sin x sin 2x Of x 5 cos 2x 2 cos x Of x 5 cos x 2 cos 2x Of x 5 sin 2x sin x Of x 2 cos 2x 5 cos x
Calculus
DifferentiationGiven vectors a 5 3 1 and b 1 3 2 calculate the following 1 3a b II a b III a x b IV proja b
Calculus
DifferentiationUsing the vectors shown in the figure write each sum or difference as a single vector A a AB BC b CD DB c DB AB D d DC CA AB C B i
Calculus
DifferentiationThe graph off is shown below For which values of x is f x zero 6 F 15 10 5 5 2 4 6 X
Calculus
DifferentiationI want to build two gardens within a rectangular field as shown below I have 800 feet of material with which to build all of the fencing for the gardens What dimensions should I use to make sure my gardens are as large as possible
Calculus
Differentiation0 0 1 0 1 0 1 0 1 0 State what the shaded area represents A Distance traveled in miles B Acceleration in miles per hour per hour C Position in miles from starting point D Final velocity in miles per hour 4 DE Next question Int s possible Velocity miles per hour Time hours
Calculus
DifferentiationDetermine whether the Mean Value Theorem can be applied to the function f x x on the closed interval 0 2 If the Mean Value Theorem f 2 f 0 be applied find all numbers c in the open interval 0 2 such that f c 2 0 MVT applies 2 3 3 O MVT applies 0 5 MVT applies 2 3 O MVT applies 1 O MVT does not apply
Calculus
Differentiation1 point a Find an angle between 0 0 and 0 360 that has the same sine as sin 152 but is not 152 0 has the same sin as sin 152 b Find an angle between 0 0 and 0 360 that has the same cosine as cos 231 but is not 231 has the same cos as cos 231 Note Do not include units in your answers Just give the numerical values 0
Calculus
DifferentiationTutorial Exercise Use the properties of summation and the Summation Formulas Theorem to evaluate the sum Use the summation capabilities of a graphing utility to verify your result Step 1 24 9i i 1 To evaluate the sum Step 2 1 9i the following property of summation can be used to find the sum n n ka k a i 1 i 1 Using the above property 9 n 24 1 24 Use the Summation Formulas Theorem to find 9 Recall that Theorem 5 2 states the following i 1 24 9 i i 1 n n 1
Calculus
DifferentiationThe voltage V in a simple electrical circuit is slowly decreasing as the battery wears out The resistance R is slowly increasing as the resistor heats up Use Ohm s law V IR to find how the current I dv dR is changing in A s at the moment when R 345 2 I 0 03 A 0 02 V s and 0 08 2 s Round your answer to six decimal places dt dt A s
Calculus
Differentiation2 5 h t NA 0 5 5 2m Using the image above find a sine or cosine equation for the function h t
Calculus
DifferentiationUse differentials to estimate the amount of tin in cm in a closed tin can with diameter 10 cm and height 18 cm if the tin is 0 04 cm thick Round your answer to two decimal place cm 3
Calculus
DifferentiationShow that the function f x y xy 8y2 is differentiable by finding values of and 2 that satisfy the following definition Definition If z f x y then f is differentiable at a b if Az can be expressed in the form Az f a b Ax fy a b Ay E AX E2Ay where and 2 0 as Ax Ay 0 0 To find an expression for Az we choose an arbitrary point a b another point arbitrarily close to the first a Ax b Ay and compute the difference between their respective f values Az f a Ax b Ay f a b a Ax b Ay 8 b Ay ab aAy b x AxAy 8b2 16bAy 8 Ay 2 ab Ay bAx AxAy but fx a b and f a b Az f a b Ax f a b Ay AxAy 8AyAy Let Ay and 2 8Ay therefore f is differentiable by the given definition AyAy SO
Calculus
DifferentiationConsider the following 1 y 6 5 4 3 2 1 0 5 1 2 1 1 1 2 3 4 a Find the least squares regression line y 5 0 5 6 X b Calculate S the sum of the squared errors Use the regression capabilities of a graphing utility to verify
Calculus
Differentiationa Approximate the area under graph a of f x 2 x over the interval 1 9 by computing the area of each rectangle to four decimal places and then adding b Approximate the area under graph b of f x 2 x over the interval 1 9 by computing the area of each rectangle to four decimal places and then adding Ay 2 1 0 Ay 2 1 1 0 1 2 3 f x 21x 5 x 27x1 3 4 5 7 6 7 80 a 9 b 9 X X Q Q C The area under graph a is approximately Round to four decimal places as needed
Calculus
DifferentiationASK YOUR TEACHER PRACTICE ANOTHE The wave heights h in the open sea depend on the speed v of the wind and the length of time t that the wind has been blowing at that speed Values of the function h f v t are recorded in feet i the table below Wave height as a function of wind speed and duration Duration hours V 10 15 20 Wind speed knots 30 40 50 60 t 5 2 4 10 15 2 4 5 7 2 20 30 8 2 2 55 8 40 54 5 What is the value of f 30 20 What is its meaning According to the table f 30 20 9 40 50 45 2 2 9 13 16 17 14 21 25 28 31 33 33 19 29 36 24 37 62 5 9 5 9 18 19 19 48 50 67 69 which means that if a 30 knot wind has been blowing in the open sea for 20 hours it will create waves with estimated heights of
Calculus
Differentiationf x x 7 Plot four points on the graph of the function the leftmost point and three additional points Then click on the graph a function butto
Calculus
Differentiation1 For this problem consider the function f x y 3y x 18yx x a Find the critical points of f
Calculus
DifferentiationThe figure shows the graph of a function At the given value of x does the function appear to be differentiable continuous but not differentiable or neither continuous nor differentiable 18 x 0 4 4 2 4
Calculus
DifferentiationA company s total cost in millions of dollars is given by C t 80 70e graph of C t is shown to the right Find each of the following a The marginal cost C t b C 0 c C 4 d Find lim C t and lim C t t a C t Do not include the symbol in your answer t where t is the time in years since the start up date The Ay 150 75 To 15 30
Calculus
DifferentiationSolve the right triangle shown in the figure Ro B A C b a DO B 33 b 50 3 A 57 a 77 5 c 92 4 A 33 a 32 7 c 92 4 A 33 a 32 7 c 42 2 A 57 a 77 5 c 60
Calculus
DifferentiationUse the given triangles to evaluate the expression Rationalize all 2 45 1 45 1 60 2 1 30 3
Calculus
DifferentiationDifferentiate f w f w SCALCET9 3 3 017 1 sec w 1 sec w 2 sec w tan w 1 sec w 12
Calculus
DifferentiationLet f x x 1 Click on the graph where the corresponding points are located for the X values of 4 3 2 1 0 1 and 2
Calculus
Differentiation3 4 018 MI Find the derivative of the function f t 7t sin t f t 7ut cos nt 7 sin
Calculus
Differentiationvalues of c in the open interval 0 8 such that f c 0 3 Let f x sin Find all values of c in the interval 1 0 such that f c 2 0 Let f x cos 4 Find 11 T 3TT
Calculus
DifferentiationSketch a graph of f x 3 2 1 5 4 3 2 1 1 2 3 4 5 1 if x 1 2x 1 if 1 x 1 if x 1 2 2 3 4
Calculus
DifferentiationExpress the number 3 3 6i in the form R cos 0 i sin 0 Rete where R 0 and 0 T NOTE Round the value of 0 to exactly two decimal places 3 3 6i
Calculus
DifferentiationUse the given graph to determine the open intervals on which the function is either increasing decreasing or constant Enter solutions using interval notation If no interval exists where the function is either increasing decreasing or constant enter DNE 10 9 8 7 6 10 9 8 7 6 5 4 3 2 1 54 3 2 2 3 4 5 597 6 7 8 9 10 3 4 5 6 7 8 9 10 The function is increasing on the interval s The function is decreasing on the interval s The function is constant on the interval s
Calculus
DifferentiationThe graph of f is given below Identify the intervals on which f is increasing or decreasing Identify the critical values and the location z values of any relative extrema of f Use interval notation to report interval solutions If multiple relative maximums or relative minimums exist enter those solutions using a comma separated list If a solution does not exist enter DNE 5 75 3 2 4 6 3 45 2 3 1 15 1 15 2 3 3 45 5 75 f is increasing on f is decreasing on The critical values of f are f has a relative maximum at x 2 3 9
Calculus
DifferentiationIn the following triangle the lengths of the sides are NOT drawn to scale and the letter Q is used to represent 8 Q 9 8 For the triangle given above express the length of s using a trigonometric function of the angle 0 Os 9 8 sin 0 Os 9 8 cos 0 Os 9 8 tan 0 Os 9 8 cot 0 Os 9 8 sec 0 8 9 8 csc 0
Calculus
Differentiation4 Grain is pouring from a chute at the rate of 10ft min forms a conical pile whose height is always twice its radius How fast is the height of the pile increasing at the instant when the pile is 2ft high
Calculus
DifferentiationIdentify the exponential function whose graph is given below Provide your answer below n 14 21 2 3